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A New Proof and an Extension of a Theorem of Millington on the Modular Group

A New Proof and an Extension of a Theorem of Millington on the Modular Group A NEW PROOF AND AN EXTENSION OF A THEOREM OF MILLINGTON ON THE MODULAR GROUP RAVI S. KULKARNI 1. Discussion and statement of the result Let F = PSL (Z) be the modular group acting canonically on the upper half plane JV. Let O be a subgroup of F of finite index d. The geometric invariants of <& include g (resp. t) the genus (resp. the number of cusps) of O\Jf , and e (resp. e ) the number 2 3 of elliptic branch points with branching index 2 (resp. 3) on <b\jV. The well-known relationship among these numbers is the Riemann-Hurwitz formula: (1.1) d=3e + 4e + \2g + 6t- 12. 2 3 A remarkable theorem of Millington [5] asserts that: (1.2) THEOREM. Given positive integers d, t and non-negative integers g, e e 2 3 satisfying (1.1), there exists a subgroup of the modular group of index d having the invariants g, t, e , e with their meanings as attached above. 2 z This theorem extends the previous work and answers a series of questions posed by H. Petersson cf. [4, 6, 7] and the references there. The method consists in relating the existence of a http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

A New Proof and an Extension of a Theorem of Millington on the Modular Group

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/17.5.458
Publisher site
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Abstract

A NEW PROOF AND AN EXTENSION OF A THEOREM OF MILLINGTON ON THE MODULAR GROUP RAVI S. KULKARNI 1. Discussion and statement of the result Let F = PSL (Z) be the modular group acting canonically on the upper half plane JV. Let O be a subgroup of F of finite index d. The geometric invariants of <& include g (resp. t) the genus (resp. the number of cusps) of O\Jf , and e (resp. e ) the number 2 3 of elliptic branch points with branching index 2 (resp. 3) on <b\jV. The well-known relationship among these numbers is the Riemann-Hurwitz formula: (1.1) d=3e + 4e + \2g + 6t- 12. 2 3 A remarkable theorem of Millington [5] asserts that: (1.2) THEOREM. Given positive integers d, t and non-negative integers g, e e 2 3 satisfying (1.1), there exists a subgroup of the modular group of index d having the invariants g, t, e , e with their meanings as attached above. 2 z This theorem extends the previous work and answers a series of questions posed by H. Petersson cf. [4, 6, 7] and the references there. The method consists in relating the existence of a

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Sep 1, 1985

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